Question

which statement is true regarding the graphed functions?

Explanation:

Step1: Analyze the intersection point

The two lines (functions \( f(x) \) and \( g(x) \)) intersect at the origin \((0,0)\) and also at \( x = -1 \)? Wait, no, looking at the graph, both lines pass through \((0,0)\) and also, let's check the intersection. Wait, actually, to find where they intersect, we can see that at \( x = 0 \), both have \( y = 0 \). Also, let's find their equations.

For \( f(x) \) (blue line): It has a slope. Let's take two points. When \( x = 0 \), \( y = 0 \); when \( x = -1 \), \( y = 3 \)? Wait, no, from the graph, when \( x = -1 \), the blue line is at \( y = 3 \)? Wait, no, the grid: each square is 1 unit. Let's take two points on \( f(x) \): (0,0) and (1, -3)? Wait, no, when \( x = 1 \), the blue line is at \( y = -3 \)? Wait, maybe better to calculate slope. From (0,0) to (1, -3): slope \( m = \frac{-3 - 0}{1 - 0} = -3 \). So equation \( f(x) = -3x \).

For \( g(x) \) (red line): Take two points: (0,0) and (3, 1.5)? Wait, no, when \( x = 3 \), \( y = 1.5 \)? No, looking at the graph, when \( x = 2 \), \( y = 0 \)? Wait, no, the red line passes through (2, 0) and (0, -1)? No, wait, the red line: when \( x = 2 \), \( y = 0 \); when \( x = 0 \), \( y = -1 \)? No, that can't be. Wait, actually, the red line passes through (0,0) and (3, 1.5)? No, maybe I made a mistake. Wait, the red line: from (0,0) to (2, 1)? Wait, no, the graph shows that at \( x = 2 \), the red line is at \( y = 1 \)? Wait, maybe the correct way is to see that at \( x = -1 \), the blue line is at \( y = 3 \), and the red line is at \( y = -1 \). Wait, perhaps the key is to find where \( f(x) = g(x) \). The intersection points are where the two lines cross. From the graph, they cross at (0,0) and also, let's check \( x = -1 \): blue line at \( x = -1 \), \( y = 3 \); red line at \( x = -1 \), \( y = -1 \)? No, that's not. Wait, maybe the question is about their intersection or values at a certain x. But since the problem is about the graphed functions, a common true statement is that they intersect at \( x = 0 \) (and maybe another point). Wait, but the main thing is to find the correct statement. For example, if the options are about \( f(2) \) and \( g(2) \), or their intersection. But since the user hasn't provided options, but the problem is about the graphed functions, the key is to analyze their intersection. Wait, the two lines intersect at \( x = 0 \) (and maybe another point). Wait, actually, looking at the graph, the two lines cross at (0,0) and also at \( x = -1 \)? No, the blue line at \( x = -1 \) is at \( y = 3 \), red line at \( x = -1 \) is at \( y = -1 \). No, that's not. Wait, maybe the correct statement is that \( f(0) = g(0) = 0 \), or that they intersect at (0,0). Alternatively, if the options are about \( f(-1) \) and \( g(-1) \), or their slopes. But since the user's problem is to find the true statement, and the graph shows that both functions pass through the origin, so \( f(0) = g(0) = 0 \). Also, when \( x = -1 \), \( f(-1) = 3 \) (since \( f(x) = -3x \), so \( f(-1) = 3 \)) and \( g(-1) = -1 \) (if \( g(x) = 0.5x \), because slope is \( 0.5 \): from (0,0) to (2,1), slope \( 0.5 \), so \( g(x) = 0.5x \)). Then \( f(-1) = 3 \), \( g(-1) = -0.5 \)? Wait, maybe I messed up the slope. Let's re-examine:

Blue line (f(x)): passes through (0,0) and (1, -3) (since from (0,0) moving right 1, down 3). So slope is -3, equation \( f(x) = -3x \).

Red line (g(x)): passes through (0,0) and (3, 1) (since from (0,0) moving right 3, up 1). So slope is \( \frac{1}{3} \)? No, that doesn't match. Wait, the red line at \( x = 2 \) is at \( y = 1 \)? No, the graph shows that the red line crosses the x-axis at (2, 0) and the y-axis at (0, -1)? No, that can't be. Wait, maybe the red line is \( g(x) = 0.5x \), because at \( x = 2 \), \( y = 1 \); at \( x = 4 \), \( y = 2 \). Yes, that makes sense. So \( g(x) = 0.5x \). Then at \( x = 0 \), both are 0. At \( x = -1 \), \( f(-1) = 3 \), \( g(-1) = -0.5 \). At \( x = 1 \), \( f(1) = -3 \), \( g(1) = 0.5 \).

But the main point is that the two functions intersect at \( x = 0 \) (and maybe another point? Wait, no, two lines intersect at one point unless they are parallel. These are not parallel (slopes -3 and 0.5), so they intersect at one point: (0,0). Wait, but the graph shows they cross at (0,0) and also, maybe I'm wrong. Wait, the blue line: when \( x = -1 \), \( y = 3 \); red line: when \( x = -1 \), \( y = -0.5 \). So they only intersect at (0,0).

So a true statement could be \( f(0) = g(0) \) (both 0), or that they intersect at (0,0), or that \( f(-1) > g(-1) \) (since 3 > -0.5), etc. But since the problem is to find the true statement, and assuming the options are about their values at a certain x or intersection, the key is to analyze the graph.

Step2: Determine the true statement

Assuming the options are like "f(0) = g(0)", "f(2) = g(2)", etc. From the graph, at x=0, both functions have y=0, so f(0) = g(0) = 0. That's a true statement.

Answer:

The functions \( f(x) \) and \( g(x) \) intersect at \( (0,0) \) (or \( f(0) = g(0) = 0 \)). (Since the problem's options are not provided, but based on the graph, the key true statement is their intersection at the origin.)